The Wikipedia article < http://en.wikipedia.org/wiki/Lévy_C_curve > says that the curve contains open sets, which seems pretty believable from the construction. It also says the Hausdorff dimension *of the boundary* is roughly 1.9340, which is consistent with the curve itself containing open set. --Dan
On Mar 10, 2015, at 6:33 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This is almost certainly a question for Gosper. Wikipedia claims that the C-curve contains open sets, and therefore has area, and has scaling dimension 2. I had been under the impression that it had a scaling dimension that was just a little less than 2.
Is this true? Suppose I have a C-curve in the complex plane with endpoints 0 and 1, with most of the bulk in the first quadrant. Where are some example solid areas? What is the total area of the object? How many differently-shaped "tiles" does it have?
On a couple of past occasions I have tried to solve the fairly hairy set of equations arising from the different ways little families of C-curve copies can intersect, in hopes of deriving the scaling dimension. I always failed -- there was too much opportunity for stupid arithmetic errors.